The Stacks project

Lemma 35.12.2. Let $\mathcal{P}$ be a property of schemes. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Assume that

  1. the property is local in the Zariski topology,

  2. for any morphism of affine schemes $S' \to S$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic, property $\mathcal{P}$ holds for $S'$ if property $\mathcal{P}$ holds for $S$, and

  3. for any surjective morphism of affine schemes $S' \to S$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic, property $\mathcal{P}$ holds for $S$ if property $\mathcal{P}$ holds for $S'$.

Then $\mathcal{P}$ is $\tau $ local on the base.


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